Characterizing the turbulent drag properties of rough surfaces with a Taylor-Couette set-up
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J. Fluid Mech. (2021), vol. 919, A45, doi:10.1017/jfm.2021.413
Characterizing the turbulent drag properties of
rough surfaces with a Taylor–Couette set-up
Pieter Berghout1, †, Pim A. Bullee1,2 , Thomas Fuchs3 , Sven Scharnowski3 ,
Christian J. Kähler3 , Daniel Chung4 , Detlef Lohse1,5
and Sander G. Huisman1, †
1 Physics of Fluids Group, Max Planck UT Center for Complex Fluid Dynamics, MESA+ Institute and
J.M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede,
The Netherlands
2 Soft Matter, Fluidics and Interfaces, MESA+ Research Institute, University of Twente, P.O. Box 217,
7500AE Enschede, The Netherlands
3 Institut für Strömungsmechanik und Aerodynamik, Universität der Bundeswehr München,
Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
4 Department of Mechanical Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia
5 Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany
(Received 11 September 2020; revised 21 February 2021; accepted 13 April 2021)
Wall roughness induces extra drag in wall-bounded turbulent flows. Mapping any given
roughness geometry to its fluid dynamic behaviour has been hampered by the lack of
accurate and direct measurements of skin-friction drag. Here, the Taylor–Couette (TC)
system provides an opportunity as it is a closed system and allows direct and reliable
measurement of the skin-friction. However, the wall curvature potentially complicates
the connection between the wall friction and the wall roughness. Here, we investigate a
highly turbulent TC flow, with a hydrodynamically fully rough, rotating inner cylinder,
while the outer cylinder is kept smooth and stationary. We carry out particle image
velocimetry (PIV) measurements in the Twente Turbulent Taylor–Couette (T3C) facility
with Reynolds numbers in the range of 4.6 × 105 < Rei < 1.77 × 106 . From these we
find, while taking into account the influence of the curved walls on the turbulence, that the
observed effects of a hydrodynamically fully rough surface are similar for TC turbulence
and flat-plate turbulent boundary layer flows (BL). Hence, the equivalent sand grain
height ks , that characterizes the drag properties of a rough surface, is similar for both
flow geometries. Next, we obtain the dependence of the torque (skin-friction drag) on the
Reynolds number for a given wall roughness, characterized by ks , and find agreement with
the same results derived from PIV measurements within 5 %. Our findings demonstrate
† Email address for correspondence: s.g.huisman@utwente.nl
© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/
licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium,
provided the original work is properly cited. 919 A45-1
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P. Berghout and others
that global torque measurements in the TC facility could be well suited to reliably deduce
wall-drag properties for any rough surface.
Key words: Taylor–Couette flow, turbulence modelling
1. Introduction
1.1. Turbulent boundary layers over fully rough walls
The transport of a fluid over a solid body or the transport of a solid body through a fluid
is always hindered by friction forces acting on the interface between the solid and the
fluid. Ideally, the solid surface is smooth, and the drag force is a purely viscous force.
In nature and engineering applications, however, solid surfaces are nearly always rough.
This means that, in addition to a modified viscous force, the roughness also results in a
pressure contribution to the drag force (‘pressure drag’), and consequently, an increase
in the total drag force (the so-called ‘drag penalty’). The contribution of the pressure
drag to the total friction drag at the surface grows with increasing roughness height.
Ultimately, when the pressure drag dominates, the surface is called hydrodynamically fully
rough.
Owing to the obvious interest in reducing the drag penalty, substantial research has
been carried out to investigate the effects of rough surfaces on wall-bounded turbulent
flows (Jiménez 2004; Flack & Schultz 2010; Chung et al. 2021). The key effect thereof
is a downward shift (by u+ ) of the mean streamwise velocity (u+ ) in the overlap (or
logarithmic) region of the turbulent boundary layer (BL) (Clauser 1954; Hama 1954). This
shift can be considered as a direct measure of the drag penalty. The mean velocity profile
for a rough wall in the overlap region is given by the Prandtl–von Kármán profile for
smooth walls, minus this shift (Pope 2000)
1
u+ = log y+ +A − u+ , (1.1)
κ
where y+ is the wall-normal distance, and the von Kármán constant κ ≈ 0.40 and A ≈
5.0 are extracted from experimental or numerical data. The superscript √ ‘+’ indicates a
normalization of the velocity u with the viscous velocity scale uτ = τw /ρ and of the
wall-normal distance y with the viscous length scale δν = ν/uτ , where τw is the wall
shear stress, ρ is the fluid density and ν is the kinematic viscosity. For a fully rough
surface, it can be derived from dimensional arguments that the velocity shift u+ depends
logarithmically on the roughness height k+ , see e.g. Raupach, Antonia & Rajagopalan
(1991) and Pope (2000). The so-called fully rough asymptote of the roughness function is
given by
1
u+ = log ks+ +A − B, (1.2)
κ
where B ≈ 8.5 is the Nikuradse constant. The equivalent sand grain roughness height ks+
is obtained by fitting, such that the velocity shift of any fully rough surface collapses with
the velocity shift of sand grains in turbulent pipe flow, which has historically grown to be
the reference case (Nikuradse 1933). Hence, the key objective in research of wall bounded
turbulent flows over rough surfaces is to relate the statistics of a rough surface to the value
of ks , which characterizes the roughness (Forooghi et al. 2017).
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Taylor–Couette flow with rough surfaces
1.2. Taylor–Couette flow
TC flow – the flow between two coaxial, independently, rotating cylinders – is a canonical
system in turbulence (Taylor 1923; Grossmann, Lohse & Sun 2016). Because the domain
is closed in all directions, global balances can be derived and monitored, which enables
extensive comparison among theory, experiments and simulations. Moreover, the torque
(corresponding to the skin friction) can be measured accurately and directly (van Gils
et al. 2012; Huisman et al. 2014), in contrast to measurements of skin friction in open
systems.
The forcing strength of the system is quantified by the ratio of the centrifugal force and
the viscous force, i.e. the Taylor number
4
1 1+η (ro − ri )2 (ri + ro )2 (ωi − ωo )2
Ta = √ . (1.3)
4 2 η ν2
Here, η is the geometric measure of curvature, namely the ratio ri /ro of the radii of the
cylinders. The subscripts i and o indicate inner cylinder and outer cylinder, respectively.
The angular velocity is denoted by ω, and ν is the kinematic viscosity.
The global response of the system is expressed as the Nusselt number Nuω , which is the
ratio between the angular velocity flux J ω in the radial direction and its laminar counterpart
ω (Eckhardt, Grossmann & Lohse 2007), as
Jlam
Jω r3 (ur ωA(r),t − ν∂r ωA(r),t )
Nuω = ω = . (1.4)
Jlam 2νri2 ro2 (ωi − ωo )/(ro2 − ri2 )
Here, ·A(r),t denotes averaging over the cylinder surface A(r) and over time t. The
Nusselt number Nuω is related to the torque T required to drive the inner cylinder. In
non-dimensional form, the torque can be expressed as
ω
T Jlam
G= = Nuω 2 . (1.5)
2πLρν 2 ν
From here on, we assume inner cylinder rotation only, hence ωo = 0, as this corresponds
to our experiments where we kept the (smooth) outer cylinder stationary at all
times.
The torque is directly related to the wall shear stress τw = T /(2πri2 L). As commonly
used in other canonical systems (e.g. the flat-plate BL), we define the friction factor Cf
following Cheng, Pullin & Samtaney (2020)
8Re2τ 4Nuω
Cf = = , (1.6)
2
Rei η(1 + η)Rei
where Rei = ri ωi d/ν, Reτ = uτ d/(2ν) and d = ro − ri .
The turbulent flow in the TC set-up is strongly influenced by the curvature of its
bounding walls, i.e. the cylinders that drive the flow. This distinguishes turbulent TC flow
from turbulent flows in other canonical systems. Bradshaw (1969) realized that the effects
of curvature on a turbulent BL are very similar to the effects of buoyancy stratification
on a turbulent BL (Obukhov 1971). In analogy to the Obukhov length (Obukhov 1971;
Monin & Yaglom 1975), he derived a length scale that separates the curved BL in a region
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P. Berghout and others
where the effects of shear dominate (i.e. production of turbulence is dominated by shear
production), and a region further away from the wall where curvature effects dominate (i.e.
the production of turbulence is dominated by curvature). For smooth wall TC turbulence,
this ‘curvature Obukhov length’ is well approximated by Berghout et al. (2020)
uτ
Lc,s = , (1.7)
κωi
where shear dominates for y 0.20Lc,s , shear and curvature effects are both significant
for 0.20Lc,s y 0.65Lc,s , and curvature effects dominate at 0.65Lc,s y. Here, y is
the distance to the cylinder, which is y = r − ri for the inner cylinder BL and y = ro − r
for the outer cylinder BL. For the derivation of Lc+ , we refer to § 3.1. Using data of the
mean velocity profiles from PIV in turbulent TC flow (Huisman et al. 2013; van der Veen
et al. 2016) and direct numerical simulations (DNS) (Ostilla-Mónico et al. 2015), the mean
angular logarithmic velocity profile in the region of the turbulent BL, where curvature
effects are important, was recently obtained for smooth wall TC flow (Berghout et al.
2020). By employing a matching argument between the velocity profiles of the turbulent
BL and the bulk region, following the work of Cheng et al. (2020), an analytical expression
for Nu(Ta) was derived (Berghout et al. 2020).
The effects of irregular boundaries (extended transverse bars in the ‘obstacle regime’, as
referred to by Jiménez (2004)) on turbulent TC flow were previously investigated by means
of experiments (Cadot et al. 1997; van den Berg et al. 2003; Verschoof et al. 2018; Zhu
et al. 2018) and DNS (Zhu, Verzicco & Lohse 2017; Zhu et al. 2018). The ratio k/d between
the height of the bars and the gap width d = ro − ri was as large as k/d = 0.05 or even
0.1. Later, Berghout et al. (2019) numerically studied the effects of sand grain roughness
(k/d = 0.019–0.087) on the turbulent TC velocity profiles, and found similar transitionally
rough behaviour as the sand grain roughness of Nikuradse (1933) in turbulent pipe flow.
However, we note that both the experimental and computational studies in TC flow suffered
from a limited scale separation between the roughness scale k and the gap width d.
In this paper, we study the effects of a hydrodynamically fully rough inner cylinder on
the turbulent wall-bounded flow, with small roughness k/d = 0.014, where k ≡ 6kσ , and
kσ is the standard deviation of the roughness elevation, following Squire et al. (2016).
In particular, we keep k much smaller than the curvature Obukhov length Lc (see (3.2)),
namely k/Lc = 0.078–0.090. We will demonstrate that to study the effects of roughness
on a turbulent flow in TC, k d is not enough. Rather, k Lc must also hold to ensure
that effects related to the streamwise curved geometry are not influencing the effects of the
roughness.
Hence, we hypothesize that effects of roughness in TC turbulence (where k Lc )
are similar to the effects of roughness in other canonical systems without streamwise
curvature. Thus, global measurements in the (closed) TC facility can be employed to
characterize drag properties of the rough surface. The outer cylinder remains smooth to
allow for optical access of the velocity profiles.
The paper is organised as follows: In § 2, we describe the experimental methods. We
then (§ 3) discuss the relevant dynamical length scales in the experiment, and elaborate
on the different regions in the BL where turbulent production is dominated by shear
effects and where effects related to the streamwise curvature of the set-up play a role.
We also comment on the scale separation and show that the roughness mainly affects
the inertial shear dominated regime, and hence, effects from the streamwise curved
geometry of the TC flow do not modify the velocity shift. In § 4, we use the mean
velocity profiles of the inner cylinder boundary layer to show that, apart from the shift, the
velocity profiles for rough and smooth inner cylinders are the same. We use this in § 5 to
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Taylor–Couette flow with rough surfaces
calculate the angular velocity shift ω+ , from which the equivalent sand grain roughness
height is determined in § 6. In § 7, we demonstrate that the bulk region of the flow is of
constant angular momentum, which is used in § 8 to obtain a relation between the torque
(skin-friction drag) and the Reynolds number for given surface roughness ks , in agreement
with our experimental results. The paper ends with a summary, conclusions and an
outlook (§ 10).
2. Experimental set-up and methods
2.1. Experimental set-up
The experiments were performed in the Twente Turbulent Taylor–Couette (T3 C) facility
(van Gils et al. 2011a), with water as the working fluid. We used a fully rough inner
cylinder with an outer radius of ri = 201.2 mm, which consisted of a smooth inner cylinder
with radius 200.0 mm and a support layer of the roughness with a thickness of 1.2 mm, and
a transparent outer cylinder with an inner radius of ro = 279.4 mm. This gave a radius ratio
of η = ri /ro = 0.720 and a gap width d = ro − ri = 78.2 mm for the rough wall cases.
For the smooth wall calculations in § 8, we used η = 0.714. The cylinders had a height of
L = 927 mm and an aspect ratio of Γ = L/d = 11.9. For inner cylinder rotation only (the
outer cylinder is stationary), the Reynolds number, Rei , is defined with the velocity of the
inner cylinder, ri ωi , and the gap width, d, as
ωi ri d
Rei = . (2.1)
ν
Using the viscous velocity, uτ , which is obtained from torque measurements, the friction
Reynolds number, Reτ , is defined as
uτ (d/2)
Reτ = . (2.2)
ν
The roughness used was P36 grit sandpaper (VSM, ceramic industrial grade), which was
fixed to the entire surface of the inner cylinder using double-sided adhesive tape (tesa
51970). We define the characteristic length scale of the roughness as k ≡ 6kσ ≈ 1.07 mm
(corresponding to the 99.8 % interval of the height), where kσ is the standard deviation of
the local roughness height h(x, y) (quantified using confocal microscopy (Bakhuis et al.
2020) on a square sample with sides of 25 mm from the rough sandpaper), and k/d =
0.014. For more details of the roughness height properties, see table 1 of Bakhuis et al.
(2020).
2.2. Experimental procedure
We performed seven experiments with different rotation rates of the inner cylinder, see
table 1. During all these experiments, the torque T that is required to drive the inner
cylinder at fixed rotational velocity was measured. The hollow reaction torque sensor
that connects the drive shaft to the middle section of the inner cylinder is indicated
in figure 1(a). By only measuring the torque on the middle section, possible end-plate
effects were eliminated (van Gils et al. 2012). During the torque measurements, PIV
was used to obtain the velocity field in the gap. To quantify the reproducibility of our
torque measurements, we compared the torque data that were captured during the PIV
experiments with three separate torque measurements thereafter. We found a spread in T
of smaller than 4 % for all cases. These direct and reproducible measurements of the torque
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P. Berghout and others
Ta (×1012 ) Rei (×106 ) Nuω Cf (×10−3 ) Reτ (×103 ) Lc+ (×103 ) kσ+ ks+ ω+ uτ (m s−1 )
0.31 0.46 312 2.21 7.6 2.6 34 189 9.31 0.21
0.57 0.62 403 2.13 10.0 3.3 44 245 10.09 0.26
0.92 0.78 513 2.14 12.7 4.1 57 317 10.80 0.33
1.47 0.99 643 2.12 16.0 5.0 72 400 11.46 0.41
2.18 1.20 784 2.12 19.5 6.0 87 484 12.06 0.50
3.55 1.54 998 2.11 24.9 7.6 111 617 12.80 0.61
4.71 1.77 1137 2.09 28.5 8.5 127 706 13.21 0.68
6.15 2.00 653 1.07 23.1 6.9 0 – – 0.58
Table 1. Control parameters, global response and relevant length scales, measured during the PIV
measurements. The Ta or Rei characterize the driving of the system. Here, Nuω is the dimensionless angular
velocity flux, Cf is the friction factor, Lc+ is the curvature Obukhov length as defined in (3.2) and kσ+
is the standard deviation of the sandpaper roughness kσ+ = (hr − hr )2 /δν , where · refers to spatial
averaging and hr is the mean roughness elevation, see Bakhuis et al. (2020). The equivalent sandgrain height
is ks+ = 5.56kσ+ . The velocity shift is ω+ and uτ is the friction velocity. The bottom row presents the values
corresponding to the smooth wall measurement of Huisman et al. (2013).
(b)
10 mm
(a)
10 mm
Laser
(c)
Sandpaper
IC
z
Camera ri
Figure 1. (a) Cross-section of the TC geometry. The tracer particles are illuminated by a 1 mm thick horizontal
laser light-sheet from the right at mid-height of the cylinders. The light scattered by the tracers is imaged from
the bottom through a mirror. The torque sensor only measures the torque of the middle part of the inner cylinder.
(b) Three-dimensional visualization of the confocal scan of the used sandpaper. (c) Cross-section of the inner
cylinder with the sandpaper attached to the surface.
(friction) have an accuracy that is comparable to the measurement accuracy of wall shear
stress in flat-plate BLs, by means of a drag balance (Baars et al. 2016).
For the PIV measurements, fluorescent polymer tracer particles (Dantec FPP-RhB-10
1–20 µm) were added to the working fluid. A horizontal laser sheet with a thickness of
approximately 1 mm illuminated the tracer particles in the working liquid at mid-height,
through the transparent outer cylinder. The laser sheet was created using a frequency
doubled Quantel EverGreen 200 mJ laser. The fluorescent light emerging from the tracer
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Taylor–Couette flow with rough surfaces
particles was imaged from below, through a window in the bottom plate of the apparatus.
For this, a 45 ◦ mirror was positioned under the bottom plate as drawn schematically in
figure 1. The camera was a high-resolution sCMOS camera (LaVision PCO.edge), with
a resolution of 2560 px × 2160 px and a pixel size of 6.5 µm. A 100 mm focal length
objective (Zeiss Makro Planar, 100 mm) was used, which gave an optical magnification of
0.17.
For each rotational velocity of the inner cylinder, 104 image pairs were acquired at
a recording frequency equal to the rotation rate. The mean velocity distribution in the
horizontal plane was computed using single-pixel ensemble correlation (Kähler, Scholz &
Ortmanns 2006; Kähler, Scharnowski & Cierpka 2012). The spatial resolution was 50 µm,
which led to approximately 1600 independent measurement points in the radial direction,
evenly spread over the entire gap. From the correlation function (obtained for every
pixel), one can directly extract the standard deviation of the velocity by integrating the
∞
probability density function σ (u) = −∞ (u − u)2 PDF(u) du, see Scharnowski, Hain &
Kähler (2012). This ensures that all turbulent scales are included in the standard deviation,
as opposed to regular PIV analysis. The velocity profiles were smoothed using a Gaussian
filter with a standard deviation of σ ≈ 0.5 mm.
3. Curvature effects, the mean velocity profile and scale separation
3.1. The relative effects of curvature and shear
To characterize and quantify the relative effects of shear and curvature in TC turbulence,
we studied the ratio S−1 , which is the ratio of turbulence production by shear over
turbulence production by curvature (Bradshaw 1969; Townsend 1976; Berghout et al.
2020)
d
uθ ur U 1 dU
−1 dr
S = = , (3.1)
1 ω dr
u uU
r θ r
where uθ and ur are the azimuthal and radial velocity fluctuations, respectively, and uθ ur
is the Reynolds stress. The mean azimuthal velocity is denoted by U and ω = U/r is the
mean angular velocity. The curvature Obukhov length Lc defined in (1.7) for a smooth
wall marks the transition from a region where the production of turbulence is dominated
by shear (y < 0.20Lc ) to a region where it is affected by curvature (y > 0.20Lc ). The
definition from (1.7) builds on the existence of a shear logarithmic region, where the
gradient of the mean angular velocity is (d/(dr))U = uτ /(κy). Once we approximate the
mean angular velocity with ω = ωi , we find that for S−1 = (uτ /(κωi y)) = 1, we have
y = Lc . Physically this means that at a distance of Lc from the wall, the shear and curvature
production of turbulence are of equal magnitude. The angular velocity scale for rough
walls is approximated as ω = ωi + ω. Thus, the generic curvature Obukhov length Lc
for smooth and rough walls can be defined with the inner cylinder rotation rate ωi and the
wall-shear stress τw only, similar to (1.7), but now for a rough wall,
uτ
Lc = , (3.2)
κ(ωi + ω)
so that Lc+ (ω+ = 0) = Lc,s+ .
Figure 2(a) presents the compensated gradient of the mean angular velocity profile
versus S−1 , calculated from the PIV results. We find fair collapse of the velocity gradients
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P. Berghout and others
(a) 6 (b) k s+ = 0
Curvature Shear and curvature Shear 101
k s+ = 189
5 k s+ = 245
k s+ = 317
r dU
U dr
4 k s+ = 400
dω+
dy+
k s+ = 484
S –1 =
y+
3 k–1 k s+ = 617
k s+ = 706
2 λ–1
100
1
100 101 10–1 100
r dU y/Lc
S –1 =
U dr
Figure 2. (a) Compensated gradient of the mean angular velocity profile versus the ratio S−1 between the
turbulence production by shear and that by curvature, see (3.1). Dotted and dashed lines represent the slope of
the logarithmic velocity profile of the shear and the curvature dominated regimes, κ −1 and λ−1 , respectively. (b)
Ratio S−1 versus the wall-normal distance shifted with the wall offset, y/Lc = (r − ri − 2kσ )/Lc , where 2kσ
is the approximated wall offset of the sandpaper. Coloured lines are calculated from the PIV data of the rough
wall cases. The grey line (kσ+ = 0) is the smooth wall profile at Ta = 6.2 × 1012 (Reτ = 23 093), obtained from
Huisman et al. (2013).
of the smooth (grey) and rough (coloured) wall profiles. When the effects of curvature
are negligible S−1 ≥ O(10), the compensated gradient of the velocity profile approaches
κ −1 ≈ 2.5. This occurs in a very small region close to the wall, where we cannot measure
owing to the presence of the sandpaper roughness. For the rough and smooth wall velocity
profiles, we find that the compensated gradient approaches λ−1 in the region where
curvature and shear affect the flow. For S−1 ≤ 1, curvature effects dominate the flow, and
a constant angular momentum region (i.e. the bulk flow) is established (Berghout et al.
2020).
3.2. The mean angular velocity profile
Figure 2(a) shows that the curvature- and shear-affected region of the BL contains
a constant compensated gradient (= λ−1 ) of the mean angular velocity. From this
observation, Berghout et al. (2020) obtained an equation of the mean angular velocity
in the shear- and curvature-affected region in the BL (in short ‘curvature log’).
The offset of the logarithmic velocity profile (with slope λ−1 ) in the curvature-
and shear-affected region, as indicated in figure 3, is a function of the wall normal
location where curvature-related effects impact the flow. From PIV results, the exact
location was found to be y+ = 0.20Lc+ , with Lc+ defined in (3.2). Therefore, the offset
is κ −1 log 0.2Lc+ + A, where A = 5.0 is the offset of the logarithmic velocity profile in the
shear-affected region (Pope 2000). The transition in the logarithmic velocity profile from
the shear-affected region to the curvature- and shear-affected region at y+ = 0.20Lc+ is
not sharp but gradual. To account for this, we introduce a constant Cbl that connects the
logarithmic velocity profiles of both regions. Berghout et al. (2020) found that A + Cbl +
(1/κ − 1/λ) log(0.2) = 1.0 for the inner cylinder, and the mean angular velocity equation
above y+ = 0.20Lc+ as
+ 1 + 1 y+ 1 + 1 1
ω = log 0.2Lc +A + Cbl + log = log y + − log Lc+ +1.0, (3.3)
κ λ 0.2Lc+ λ κ λ
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Taylor–Couette flow with rough surfaces
Viscous Shear Shear and curvature Curvature
ω+
Smooth
1 log 0.2L+ + A 1
k c
λ
1
ω+
k Rough
1 log 0.2L+ + A – ω+
k c Bulk
A
Boundary layer
y+ = 0.20L+c y+ = 0.65L+c log y+
Figure 3. Schematic of the various regions in smooth and IC-rough turbulent TC flows at matched Lc+ . The
height y+ = 0.65Lc+ is defined as the location where the logarithmic profile with slope λ−1 ends and the
constant angular momentum region of the bulk velocity starts.
with Cbl = −3.30. The transition from the curvature- and shear-affected region to the
constant angular momentum region occurs at y+ = 0.65Lc+ . We take this height as our
definition of the boundary layer height, above which is the bulk region of constant angular
momentum.
3.3. Scale separation
Key to the understanding of the effects of roughness in TC turbulence is the
concept of scale separation. To illustrate this, in figure 2(b), we plot S−1 versus the
wall-normal distance y/Lc = (r − ri − 2kσ )/Lc . We note that the wall offset (or zero plane
displacement) 2kσ+ of the rough wall is an approximation. Because we focus on the flow
region that is relatively far away from the wall, and ks /d 1, the choice of the zero plane
displacement hv has negligible influence on the results, as far as 0 < hv < k (Raupach
et al. 1991; Squire et al. 2016). To quantify this, we set hv = 0 and h0 = 6kσ (valley and
peak of the roughness) and find that ω+ only varies by 4 %. As a reference, we also plot
the smooth wall profile (grey) at Ta = 6.2 × 1012 (Huisman et al. 2013), together with the
rough wall profiles (colours).
Table 1 presents the relevant dynamical length scales in the experiments: namely, Reτ ,
Lc+ and kσ+ . The friction Reynolds number Reτ from (2.2) gives the ratio of the largest
dynamical length scale in the TC set-up to the viscous length scale δν . Here, Reτ is of
the same order as in the smooth wall TC experiments by Huisman et al. (2013), where it
was Reτ = 488–23 093, which is comparable to the rough BL experiments by Squire et al.
(2016), where Reτ = 2890–29 900.
The roughness scale in our experiments is much larger than the viscous length scale
δν , i.e. kσ+ = 34–127 1 (ks+ = 188–704 1), and thus pressure drag dominates over
viscous drag. For the flat-plate BL experiments of Squire et al. (2016) in the fully rough
regime, we estimate that kσ+ = 9–12 is required, based on the data for which U + > 8.0
in Squire et al. (2016). Hence, we are confident that we are indeed far in the fully rough
regime. We also find that the roughness sublayer height ≈ (6 − 9)kσ+ (approximately
0.5 times the roughness spacing Chung et al. 2021) is smaller than the outer bound
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P. Berghout and others
of the shear dominated logarithmic region, ≈ 0.2Lc+ . For the lowest roughness, we
have 9kσ+ /0.2Lc+ = 0.60, and for the highest roughness, it is 9kσ+ /0.2Lc+ = 0.66. This
separation of length scales allows for a region where the logarithmic velocity profile can
form, albeit marginally. For example, in the smooth wall experiments of Huisman et al.
(2013), such a profile was found between 50 ≤ y+ ≤ 600 for comparable Ta. We finally
find that the outer bound of the curvature dominated logarithmic region Lc+ is smaller than
the outer length scale Reτ , so that 0.65Lc+ /Reτ ≈ 0.33. For y+ > 0.65Lc+ , the curvature
dominated bulk, a constant angular momentum region forms. The occurrence and extent of
this constant angular momentum region depends on the radius ratio η, i.e. Lc+ /Reτ depends
on η.
From table 1, we find that δν kσ < 0.20Lc , hence the roughness only affects the
inertial region where curvature effects are negligible. We therefore expect that the velocity
shift of that region is similar to that of identical sandpaper in a flat-plate turbulent BL.
Therefore, we would expect a fully rough asymptote with slope κ −1 and a similar value of
ks as we would measure for identical sandpaper in a flat-plate turbulent BL.
4. Mean velocity profiles of the inner cylinder boundary layer
In this paper, we will show the angular velocity profile ω+ ( y+ ) rather than the azimuthal
velocity profile u+ ( y+ ), as it is ω+ ( y+ ) that is expected, given the arguments based on the
Navier–Stokes equations, to have a logarithmic profile (Grossmann, Lohse & Sun 2014).
Figure 4(a) shows the angular velocity profiles over the rough wall ω+ = ωi −
ω(r)t /ωτ , with ωτ = uτ /ri , versus the wall-normal coordinate y+ . In this and the next
section, we focus our analysis on the mean velocity profiles of the inner cylinder BL, hence
uτ = uτ,i throughout. In § 7, we will report on the bulk profiles. We refer to Berghout et al.
(2020) for an analysis of the smooth velocity profiles of the outer cylinder BL.
Figure 4(a) shows that with increasing viscous scaled roughness height, the rough
wall profiles are increasingly shifted downwards, as expected. More importantly, we find
from the diagnostic function y+ (dω+ /dy+ ) (a useful representation of the compensated
gradients (Pope 2000)) in figure 4(b) that the slope λ−1 of the shear- and curvature-affected
logarithmic region is the same for rough wall TC turbulence as for smooth wall TC
turbulence (grey line). Unfortunately, we could not resolve the very thin spatial region
where a shear-dominated logarithmic was found by Huisman et al. (2013), as the roughness
peaks obstruct the view for the PIV very close to the wall. Note that the range of y+ /Lc+ in
figure 4(b) is slightly shorter than in figure 5(a), because we employ a Gaussian smoothing
before calculating the gradient of the velocity.
For a rough wall, ω+ is a function of the equivalent sand grain roughness height
ks and the curvature length Lc , so that ω+ (ks+ , ks /Lc ). When ks Lc , the angular
+
velocity shift only depends on ks , and the shift becomes ω+ (ks+ ). Because the inner
cylinder rotates, the plus sign in the denominator of (3.2) is connected with the increase
of angular fluid velocity in the inner cylinder BL owing to the roughness. When we
normalize the wall-normal distance with Lc+ , we expect the transition from a curvature
logarithmic velocity profile to the constant angular momentum bulk velocity profile to
occur at y+ /Lc+ = 0.65. In figure 4(b), we find a fair collapse of both smooth and rough
wall profiles in the wall-normal direction, when normalized with curvature length Lc+ . The
slopes of the curvature-affected logarithmic region of all profiles, rough and smooth, in the
domain 0.20Lc+ < y+ < 0.65Lc+ , fall in the range λ = 0.60–0.65.
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Taylor–Couette flow with rough surfaces
(a) 30 (b) 5
k s+ = 0 k s+ = 400
k s+ = 189 k s+ = 484
4
k s+ = 245 k s+ = 617
20 k s+ = 317 k s+ = 706
3
dω+
dy+
ω+
y+
2
10
1
Dashed lines: Smooth (theoretical)
Solid lines: PIV
0 0
102 103 104 0.20 0.65 0
10–1 10
y+ y+/L+c
Figure 4. (a) Mean angular velocity ω+ versus the wall-normal distance y+ . The solid lines are the measured
rough wall profiles. The dashed lines represent the theoretical smooth wall reference profiles (colours are the
same), calculated from (3.3) and at matching Lc+ and Reτ with the rough wall profiles. (b) The compensated
gradient of the rough wall profiles in (a), where the wall-normal distance is normalized with the curvature
length Lc+ . The colours are the same in both figures. The grey line is the smooth wall profile at Ta = 6.2 × 1012 ,
obtained from Huisman et al. (2013). The dashed horizontal line represents the slope λ−1 of the logarithmic
velocity profile in the region where turbulence production is governed by effects of curvature and shear. The
dotted horizontal line represents the slope κ −1 of the logarithmic velocity profile in the region where turbulence
production is dominated by shear.
(a) (b) 20
ω+ = 1/0.40 log ks+ + 5.0–8.5
12 15 ω+ ∝ 1/0.64 log ks+
ω+ 10 10
5
8
ks = 5.54kσ = 0.97 mm
0
10–2 10–1 100 102 103
y+/L+c ks+
Figure 5. Velocity shift ω+ of the rough wall profiles with respect to the reference smooth wall profiles.
(a) Velocity shift versus the wall-normal distance y+ /Lc+ (colours are the same as figure 4). (b) The velocity
shift ω+ , crosses in (a), versus the equivalent sand grain height ks+ . Black symbols are the experimental
values. The solid black line is the fully rough asymptote of Nikuradse (1933), (5.1). The solid blue line is an
illustration of the curvature fully rough asymptote, with slope λ−1 and arbitrary vertical shift.
5. The fully rough asymptote
From the observation that both smooth and rough wall velocity profiles possess the same
slope λ−1 of the curvature logarithmic region (figure 4b), we proceed to calculate the
angular velocity shift ω+ . The major advantage of the TC set-up is that the friction
can be measured in a straightforward manner by measuring the torque. Hence, we can
directly obtain the velocity shift by subtracting the rough velocity profile from the smooth
velocity profile, rather than by means of fitting a velocity profile first to obtain uτ , e.g. the
Clauser fit. Owing to the roughness, the angular velocity profiles in the shear logarithmic
region are shifted, as discussed in § 3.1 and illustrated in figure 3. This shift remains also
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P. Berghout and others
in the curvature logarithmic region, where we will now quantify it. The offset of that
region scales with (1/κ) log(0.20Lc+ ) + A (figure 3). Hence, it is imperative to calculate
the angular velocity shift from the smooth wall velocity profile at matching Lc+ .
Figure 4(a) shows these smooth wall profiles (dashed), where the colours match the
respective rough wall cases, and both Lc+ and Reτ are matched. The velocity shift
ω+ ( y+ ) from the theoretical smooth wall profile, (3.3), is plotted in figure 5(a). The
horizontal plateaus confirm the similarity of the slopes of the velocity profiles. We extract
ω+ at y+ ≈ 0.4Lc+ and plot the shift versus the roughness height in figure 5(b). When we
fit a function of the form ω+ = (1/a) log k+ + b through all seven data points, we obtain
a = 0.34 ± 0.02 to within 13 % of the von Kármán constant κ = 0.39, as measured in TC
turbulence (Huisman et al. 2013) as the slope of the shear dominated logarithmic profile.
This confirms our hypothesis, as discussed in § 3.1, where the fully rough asymptote for
δν k < Lc has slope κ −1 . For reference, this is much higher than λ−1 ≈ 0.64−1 , blue
line in figure 5(b).
To obtain a measure of the equivalent sand grain roughness height ks , we fit the data
points to the fully rough asymptote of Nikuradse (1933), with κ ≈ 0.40, which is in
accordance with the value reported for the fully rough asymptote (Jiménez 2004),
1
ω+ (ks+ ) = log ks+ +5.0 − 8.5, (5.1)
κ
and obtain ks = 5.54kσ = 0.97 mm. For reference, the typical grain size is estimated by
6kσ = 1.05 mm (Bakhuis et al. 2020).
6. The equivalent sand grain roughness height
The hypothesis in this research, postulated in § 3.1, is that the fully rough asymptote in TC
turbulence with δν k < Lc is the same (or very similar) to the fully rough asymptote
in flows without streamwise curvature. We have already demonstrated in § 5 that the
slope κ −1 of the fully rough asymptote is indeed (almost) the same. This leaves us with
a comparison of the value of ks between TC turbulence and canonical systems without
streamwise curvature.
In the literature, we have found two reports on turbulent flows over sandpaper roughness:
the work of Squire et al. (2016), which employed 36 grit sandpaper in a turbulent BL, and
Flack et al. (2007), who employed (12, 24, and 80) grit sandpaper in turbulent BL flow.
In the rough wall TC experiments reported here, we used grit 36 sandpaper. However, it is
essential to realize that sandpaper is not only defined by the grit size. Other statistics, such
as the skewness (an important parameter (Forooghi et al. 2017), which is 0.93 here and
only 0.09 in the paper by Squire et al. (2016)), do vary with manufacturing methods. We
have tried to use the very same sandpaper type (SP40F, Awuko Abrasives) as Squire et al.
(2016). Unfortunately, the sandpaper turned out not to be waterproof, and detached from
the inner cylinder. We then applied new water-resistant sandpaper (VSM, P36 grit ceramic
industrial grade), with different surface roughness statistics.
To compare the drag property of the sandpaper surfaces in TC to the respective
sandpaper surfaces in literature, we plotted the relationship between ks and the
root-mean-square height and skewness in figure 6. The surface properties of the sandpaper
surface from Flack et al. (2007) are taken from Flack & Schultz (2010). The solid black
line is the empirical correlation from Flack et al. (2020). We find that the relation between
ks and the skewness Sk and the root-mean-square height kσ of the sandpaper used in our
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Taylor–Couette flow with rough surfaces
40
Flack et al. (2007)
Squire et al. (2016)
30 Gul et al. (2021) ES
Taylor–Couette (2020) Flack et al. (2007) > 0.35
ks/krms
20
ks/krms = 2.48(1 + Sk)2.24
Squire et al. (2016) 0.482
Present 1.45
10
0 0.5 1.0 1.5 2.0
Sk
Figure 6. Relationship between the equivalent sand grain roughness height divided by the root-mean-square
height ks /kσ , and the skewness parameter Sk of different sandpaper surfaces. The solid black line is the
empirical correlation for Sk > 0 from Flack et al. (2020). Data from turbulent boundary layer flow using grit
(12, 24 and 80) sandpaper (Flack, Schultz & Connelly 2007), of which the surface statistics are listed by Flack
& Schultz (2010), turbulent boundary layer using grit 36 (Gul & Ganapathisubramani 2021) and (Squire et al.
2016), and turbulent TC flow using grit 36 (present). ES is the effective slope defined as |dk/d(rθ )| (Napoli,
Armenio & De Marchis 2008).
rough wall TC experiments is consistent with the empirical trend given for the sandpaper
used in rough wall turbulent BL flow analysis. It was hypothesized by Chung et al.
(2021) that for fixed skewness, ks /k versus ES increases with increasing ES, exhibits
a maximum, and then will decrease with further increasing ES. The results in figure 6
support this hypothesis. Whether the deviation originates from the difference between TC
and canonical systems without curvature, or originates from the different surface statistics
(e.g. the ES for the present surface is higher, which indicates a denser surface), remains to
be resolved.
7. The constant angular momentum region in the bulk
Thus far, we have discussed the velocity profiles of the inner cylinder boundary layer,
i.e. y+ < 0.65Lc+ . By means of matching this profile to the bulk velocity profile at the
boundary layer height, one can derive the relationship between the torque Nuω (Ta) and the
velocity of the inner cylinder (Berghout et al. 2020; Cheng et al. 2020). For smooth wall
inner cylinder rotating turbulent TC flow, it is well known that the angular momentum in
the bulk (Mb ) is constant (Wendt 1933; Townsend 1976), and, in fact, very close to half
the inner cylinder angular momentum (Mi = ωi ri2 ), Mb = 0.5Mi , for a stationary outer
cylinder. For rough wall TC flow however, and especially for asymmetric roughness when
the inner cylinder is of a different roughness height than the outer cylinder, the exact
value of Mb is a priori unknown. However, what was shown is that for very rough walls,
the bulk azimuthal velocity profile is shifted towards the rough cylinder (Zhu et al. 2017,
2018; Berghout et al. 2019).
If the bulk region velocity conforms to a constant angular momentum, it should
match the angular momentum at the edge of the BL r = ri + δr , where δr = 0.65Lc .
The momentum ratio (Mb /Mi ) is the angular momentum in the bulk over the angular
momentum of the inner cylinder
Mb ω|y=δr (ri + δr )2
= , (7.1)
Mi ωi ri2
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P. Berghout and others
(a) 1.0 k s+ = 0 k s+ = 400 (b)
k s+ = 189 k s+ = 484 0.7
0.8
k s+ = 245 k s+ = 617
k s+ = 317 k s+ = 706
0.6
M/Mi
ω/ωi
0.6
0.4
0.2 0.5
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
(r – ri)/d (r – ri)/d
Figure 7. Bulk velocity profiles. (a) The mean angular velocity normalized with the inner velocity ω/ωi versus
the radius (r − ri )/d normalized with the gap width d. The profiles for different roughness heights ks+ are
compared. The bulk profile is strongly shifted towards the rough inner cylinder, as the roughness there enhances
the coupling between the inner BL and the bulk, similarly as the ribs have done in Zhu et al. (2018). (b) The
angular momentum M normalized with the inner cylinder angular momentum Mi . Solid lines are the PIV results
and dashed lines (Mb /Mi ) are calculated from (7.1). The colours are the same in both figures. The grey line is
the smooth wall profile at Ta = 6.2 × 1012 , obtained from Huisman et al. (2013).
where ω|y=δr = ωτ,i (ωi+ − ωr+ ( y+ = δr+ )), and we use the velocity profile of the rough
inner cylinder BL, figure 3 and (3.3), ω+ , ωr+ ( y+ = δr+ ) = (1/λ) log δr+ + (1/κ −
+ + 1.0 − ω+ . Figure 7 compares the result from (7.1) (dashed line) with
1/λ) log Lc,r
the experimentally obtained velocity profiles (solid lines), which demonstrates agreement
between the calculated and the measured profiles. This supports the assumption that the
rough-wall velocity profiles also conform to a constant angular momentum in the bulk.
Finally, we point out that the ‘undershooting’ and ‘overshooting’ of the profiles in the
bulk, i.e. the slight increase in M with increasing r, is likely an effect of the turbulent
Taylor vortices, and is therefore expected to depend on the height coordinate z (Huisman
et al. 2014). This arises from the detaching plumes which are transported to the other
side of the gap by the Taylor rolls. Similar overshooting is well known from temperature
profiles in turbulent Rayleigh–Bénard flow (Tilgner, Belmonte & Libchaber 1993; Ahlers,
Grossmann & Lohse 2009).
8. Calculation of Nuω (Ta) and Cf (Re)
Because the angular momentum in the bulk is, to a good approximation, constant, we
can match the angular momentum of the inner cylinder BL at the BL height with the
angular momentum of the outer cylinder BL at the BL height, i.e. M(δi,r ) = M(δo,s ). This
approach is based on the matching of the BL and bulk velocity profiles in the recent CPS
model (Cheng et al. 2020). Subscripts (i, o) refer to inner cylinder and outer cylinder
BL quantities, where subscripts (s, r) refer to smooth and rough wall quantities, and δ =
+ +
0.65Lc for the inner cylinder and outer cylinder (rough and smooth) so that δi,r = αLc,i,r ,
+ +
δo,s = αLc,o,s with α = 0.65. The matching argument becomes
+ + + +
(ri + δi,r )2 ωτ,i ωIC (δi,r ) = (ro − δo,s )2 ωτ,o ωOC (δo,s ), (8.1)
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Taylor–Couette flow with rough surfaces
where we realize that ωτ,o = η2 ωτ,i . We substitute the BL equations for respectively rough
and smooth walls into (8.1) and obtain
+ 1 + 1 1 + +
(ri + δi,r ) ωτ,i ωi − log(δi,r ) −
2
− log(Lc,i,r ) − Ci + ω
λ κ λ
1 + 1 1 +
= (ro − δo,s ) ωτ,o
2
log(δo,s ) + − log(Lc,o,s ) + Co . (8.2)
λ κ λ
+
The rough wall, inner cylinder BL height δi,r and the velocity shift ω+ are functions of
the sand grain size ks+ . This makes the matching equation more involved, compared with
the smooth wall case (Berghout et al. 2020; Cheng et al. 2020).
Following Cheng et al. (2020), we now rewrite the equation in terms of Reτ,i and Rei .
The inner cylinder angular velocity becomes
Rei
ωi+ = . (8.3)
2Reτ,i
The equivalent sand grand size is
ks
ks+ =2 Reτ,i = Reτ,i . (8.4)
d
The fully rough asymptote from (5.1) can now be rewritten as
1
ω+ = log(Reτ,i ) + A − B. (8.5)
κ
+ + +
The inner cylinder, rough wall, BL height δi,r is rewritten from δi,r = αLc,i,r as
+ 2αηReτ,i Rei 1
δi,r = ; with Z = + log(Reτ,i ) + A − B . (8.6)
κ(1 − η)Z 2Reτ,i κ
+ is rewritten from δ + = αL+
The outer cylinder, smooth wall, BL height δo,s o,r c,o,r as
+
4αη2 Re2τ,i
δo,s = , (8.7)
κ(1 − η)Rei
We can now substitute (8.3)–(8.7) into (8.2), and obtain
α 2 Rei 1 2αηReτ,i
1+ − log
κZ 2Reτ,i λ κ(1 − η)Z
1 1 2ηReτ,i 1
− − log + log(Reτ,i ) + A − B − Ci
κ λ κ(1 − η)Z κ
2
2αηReτ,i 1 4αη Re2τ,i
2
1 1 4η2 Re2τ,i
= 1− log + − log +Co .
κRei λ κ(1 − η)Rei κ λ κ(1 − η)Rei
(8.8)
This implicit equation can be solved numerically to obtain Reτ,i (Rei ) with parameters
Ci = 1.0, Co = 2.0, A = 5.0, B = 8.5, κ = 0.39 (in accordance with Huisman et al.
2013; Berghout et al. 2020), λ = 0.64, α = 0.65 for these experiments, η = 0.714 and
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P. Berghout and others
(a) 104 (b) 10–2
Smooth
Rough
103
Nu Cf
102
10–3
101
1010 1012 1014 105 106 107
Ta Rei
Figure 8. Global response of the rough ks /d = 0.012 and smooth wall TC turbulence. (a) Nusselt number
Nuω = (2Re2τ η(1 + η))/Rei versus Ta. (b) Friction factor Cf = 8Re2τ /Re2i versus Rei . Blue diamonds are the
smooth wall experiments of van Gils et al. (2011b), where the solid blue line shows the theory of Berghout
et al. (2020) for smooth wall TC turbulence. Black squares are the rough inner cylinder measurements from the
present work. The solid black line is (8.8).
= 2(0.9694/78.2) = 0.024. Finally, by means of (1.3)–(2.1), we express the result
Reτ,i (Rei ) into Nuω (Ta) and Cf (Rei ), respectively.
Figure 8 presents the final result, together with the experimental data from smooth
walls (van Gils et al. 2011b) and with the equation for smooth wall TC (Berghout et al.
2020) (grey). The black open squares represent the fully rough inner cylinder rotating TC
experiments presently. The black solid line is our calculation from (8.8). We emphasize
that no fitting parameters were used. All parameters find their origin in the velocity
profiles, and originate from the slopes of the logarithmic velocity profiles (κ −1 , λ−1 ),
the offset of the smooth velocity profile (A, Ci , Co ) or the BL thickness fit for smooth
walls α. This reflects that all parameters are universal for all radius ratios, and cannot
and need not be ‘tuned’. Regarding the quantification of the accuracy and repeatability
of the torque measurements, we carried out 6 individual time series measurements of
the torque at a fixed Re = 0.99 × 106 . We find that the mean standard deviation of the
individual time series of the torque, binned over 1 min intervals, was 1.72, which is larger
than the error in the torque sensor typically expressed by manufacturers as a percentage
of the maximum rated load. The maximum spread in the mean Nusselt numbers among
the different measurements was 5.54. Both numbers were smaller than the marker size in
figure 8(a).
The agreement between (8.8) and the experimental data is convincing. The mean
absolute error was only 4.9 %, and for the three highest Taylor numbers, the error was less
than 2 %. This implies that from straightforward measurements of the torque, for given
inner cylinder rotation speed, we can calculate the value of ks with a reasonable accuracy.
This means that the TC facility can potentially be used for direct and fast measurements
of surface drag properties, as characterized by ks .
9. Discussion
In the foregoing sections, we have shown that the TC set-up can be employed to study the
drag of rough surfaces. In particular, we have investigated the necessary scale separation,
the functional form of the velocity profiles and, from the velocity profiles, the calculation
of Nuω (Ta). Here we address some final implications of the experiments and modelling
ideas.
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Taylor–Couette flow with rough surfaces
log(y/d)
Lc δν, Lc k
Gap d/d = 1
Constant ang. mom.
Curv
atur
e log
She
ar l
og Lc/d ∝ 1/Z
Roughness k/d = constant
δν/d = 1/Reτ
log(Reτ)
Smooth Transitionally Fully Obstacle
rough rough
Figure 9. Schematic of the various rough TC flow regimes and their relationships with the mean velocity
profile.
The calculation of Nuω (Ta) mostly follows the idea of matching velocity profiles
of Cheng et al. (2020). However, Cheng et al. (2020) assumed that the bulk angular
momentum is always 0.5 times the inner cylinder angular momentum. In § 7, we have
shown that this is not the case when the drag of the inner cylinder is different from the
drag of the outer cylinder. Consequently, we matched the angular momentum of the IC BL
with the bulk at the boundary layer height, and calculated the bulk angular momentum as
a function of the roughness height (see figure 7 and (7.1)).
In § 5, we showed that the rough IC BL velocity profile follows, to a good approximation,
the fully rough asymptote. Hence, we conclude that the turbulent flow occupies the fully
rough regime. However, the friction factor Cf does not exhibit fully rough behaviour, i.e.
Cf (Rei ) is not constant, see figure 8(b). This arises from the fact that only the IC is rough
and the outer cylinder is still smooth. Much like the experiments of Zhu et al. (2018), who
found that only when both cylinders are rough and pressure drag dominates viscous drag
at both cylinders is τw ∝ Re2i , i.e. Cf (Rei ) = Constant. If both cylinders would be rough,
we could modify the right hand side of (8.2) and match two rough angular momentum
profiles. This is left for future research.
Finally, we note that a limit exists for which the modelling, as discussed in this paper,
would no longer work. At that limit, the dispersive fluctuations of the rough wall are not
constrained to the shear logarithmic region, i.e. hr /(0.20Lc ) > 1 (with hr as the roughness
sublayer height) and the flow enters the obstacle regime, see figure 9. However, we realize
that this limit is far away in terms of Reτ , as Lc /d ∝ 1/Z ∝ 1/(Rei /Reτ + log(Reτ )). For
the current system configuration, we calculate that hr > 0.20Lc for Reτ > O(107 ), with
hr ≈ 2.5ks , see Berghout et al. (2019). In fact, the obstacle regime might itself consist
of two varying regimes. One where 0.20Lc < hr < Lc , and the roughness penetrates the
curvature logarithmic region. Then a second regime where hr > Lc and the roughness
penetrates the bulk.
10. Summary, conclusions and outlook
We conducted experiments of inner cylinder rotating (and stationary outer cylinder)
Taylor–Couette (TC) turbulence with a rough inner cylinder and a smooth outer cylinder.
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P. Berghout and others
We measured the torque and, by means of PIV, the mean angular velocity profiles.
The rough surface consisted of P36 industrial grade sandpaper, where the roughness
height k = 6kσ , with kσ as the standard deviation of the roughness height, over the gap
width d was k/d = 0.014. The roughness height k was much larger than the viscous
length scale δν , such that k/δν = 204–762. The velocity shift of the rough wall angular
velocity profiles, compared with the reference smooth wall, in the ‘log-law region’, was
ω+ > 9 over the whole range of 4.6 × 105 < Rei < 1.77 × 106 . Hence, the sandpaper
was hydrodynamically fully rough. Furthermore, the roughness height k < 0.090Lc ,
where 0.20Lc is the height (Berghout et al. 2020) that separates the region in the BL
where production of turbulence is dominated by shear from the region in the BL where
production of turbulence is affected by streamwise curvature.
Using the mean azimuthal velocity profiles, we found that the slope of the fully rough
asymptote, characterized by κ = 0.34 ± 0.02, was similar to previous findings in flat-plate
BLs κ ≈ 0.38. Also, the value of the equivalent sand grain roughness height ks compared
reasonably well with those found for sandpaper in flat plate BLs (Flack et al. 2007; Squire
et al. 2016).
Finally, to obtain the relationship between the dimensionless torque and dimensionless
driving of the system Nuω (Ta), we employed a matching argument between the inner
cylinder BL rough mean angular momentum profile at the inner cylinder BL height, and
the smooth outer cylinder BL mean angular momentum profile at the outer cylinder BL
height, based on the CPS model of Cheng et al. (2020), see also Berghout et al. (2020).
To justify this, we first showed that for a rough wall inner cylinder, a region of constant
angular momentum exists in the bulk. We find a convincing overlap between the calculated
value of the torque (or wall shear stress), and the experimentally measured values of the
torque, with a mean absolute error of 4.9 %.
These findings indicate that the turbulent TC facility can be a valuable set-up
for characterizing the turbulent drag properties of any rough surface. Direct and
straightforward measurements of the torque can now be translated to a value of the
equivalent sand grain roughness height ks . It seems that the value of ks found in TC is
similar to the value of ks found in flat-plate BLs.
As an outlook to future work, we propose that more studies in both turbulent flat-plate
BLs and turbulent TC flow, with identical rough surfaces, are carried out to further
compare the drag properties of these surfaces. Further unanswered questions include
the effects of even more considerable roughness penetrating the curvature-affected
logarithmic regime of the BL, which is related to finding the slope of the fully rough
asymptote in that region. This could also be achieved by employing a TC set-up with a
lower radius ratio η, thus increasing curvature effects.
Acknowledgements. We would like to thank B. Benschop, M. Bos and G.W. Bruggert for their technical
support, Y.A. Lee for his support in the lab, and M.A. Bruning for discussions. We also thank D. Pullin for his
generous and profound comments on the manuscript.
Funding. This study was funded by the Netherlands Organisation for Scientific Research (NWO) through
the Multiscale Catalytic Energy Conversion (MCEC) research center and the GasDrive project 14504, by
the European Research Council (ERC) Advanced Grant ‘Droplet Diffusive Dynamics’, and by the Priority
Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungs-gemeinschaft.
Declaration of interests. The authors report no conflicts of interest.
Author ORCIDs.
Pieter Berghout https://orcid.org/0000-0002-7231-3132;
Pim A. Bullee https://orcid.org/0000-0001-7602-7726;
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